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Systems & Control: Foundations & Applications
Series Editor
Tamer Ba§ar, University
of
Dlinois at Urbana-Champaign
Editorial Board
o
Karl
lohan
AstrOID,
Lund Institute
of
Technology, Lund, Sweden
Han-Fu Chen, Academia Sinica, Beijing
William Helton, University
of
California, San Diego
Alberto Isidori, University
of
Rome (Italy) and
Washington University, St. Louis
Petar
V.
Kokotovic, University
of
California, Santa Barbara
Alexander Kurzhanski, Russian Academy
of
Sciences, Moscow
and University
of
California, Berkeley
H. Vincent Poor, Princeton University
Mete Soner,
K~
University, Istanbul
Daniel
Liberzon
Switching in
Systems and Control
Birkhauser
Boston -
Basel-
Berlin
Daniel Liberzon
Coordinated Science Laboratory
University ofllIinois
at
Urbana-Champaign
Urbana,
II...
61801
U.S.A.
Library
of
Congress Cataloging-in-Publication
Data
Liberzon, Daniel, 1973
Switching in systems and control!Daniel Liberzon.
p. em. - (Systems and control)
Includes bibliographical references and index.
ISBN
0~8176-4297-8
(alk. paper) - ISBN 3-7643-4297-8 (alk. paper)
1.
Switching theory.
2.
Automatic control. I. Title.
II.
Series.
TK7868.S9L53 2003
621.3815'372-dc21 2003050211
CIP
AMS Subject Classifications:
Primary: 93B12, 93D05, 93D09, 93D15, 93D20; Secondary: 34D20,
34D30, 34H05
Printed on acid-free
paper
<Q2003
Birkhliuser Boston
~®
Birkhiiuser
Iq)l)
All rights reserved. This work may not be translated
or
copied in whole
or
in partwithout the written
permission
of
the publisher
(BirWuser
Boston, cia
Springer~Verlag
New York, Inc., 175 Fifth
Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form
of
information storage and retrieval, electronic
adaptation, computer software,
or
by similar
or
dissimilar methodology now known
or
hereafter
developed is forbidden.
The use in this publication
of
trade names, trademarks, service marks and similar terms, even
if
they
are not identified as such, is not to
be
taken as an expression
of
opinion as to whether
or
not they are
subject
to proprietary rights.
ISBN 0-8176-4297-8 SPIN 10888361
ISBN 3-7643-4297-8
Typeset by the author.
Printed in the United States
of
America.
987654321
Birkhauser
Boston·
Basel·
Berlin
A member
of
BertelsmannSpringer Science+Business
Media
GmbH
Contents
Preface
I
Introd
uction
1
Basic
Concepts
1.1
Classes of hybrid
and
switched
systems.
1.1.1 State-dependent switching . . . .
1.1.2 Time-dependent switching . . . .
1.1.3 Autonomous
and
controlled switching
1.2 Solutions of switched systems
.....
1.2.1
Ordinary
differential equations
1.2.2 Zeno
behavior.
. . .
1.2.3 Sliding modes
....
1.2.4 Hysteresis switching
II
Stability
of
Switched
Systems
2
Stability
under
Arbitrary
Switching
2.1 Uniform stability
and
common Lyapunov functions
2.1.1 Uniform stability concepts
...
2.1.2 Common Lyapunov functions .
2.1.3 A converse Lyapunov theorem
2.1.4 Switched linear systems . . . .
ix
1
3
3
5
6
8
9
9
10
12
14
17
21
21
21
22
24
26
vi
Contents
2.1.5 A counterexample .
2.2 Commutation relations and stability
2.2.1 Commuting systems
.....
2.2.2 Nilpotent and solvable Lie
algebras.
2.2.3 More general Lie algebras . . . . . .
2.2.4 Discussion of Lie-algebraic stability criteria
2.3 Systems with special structure
2.3.1 Triangular systems
....
2.3.2 Feedback
systems.
. . . .
2.3.3 Two-dimensional systems
3
Stability
under
Constrained
Switching
3.1 . Multiple Lyapunov functions .
3.2 Stability under slow switching .
3.2.1 Dwell time .
3.2.2 Average dwell time .
3.3 Stability under state-dependent switching
3.4 Stabilization by state-dependent switching .
3.4.1 Stable convex combinations . .
3.4.2 Unstable convex combinations
III
Switching
Control
28
30
30
34
37
41
42
43
45
51
53
53
56
56
58
61
65
65
68
73
4
Systems
Not
Stabilizable
by
Continuous
Feedback
77
4.1 Obstructions to continuous stabilization
77
4.1.1 State-space
obstacles.
77
4.1.2 Brockett's condition . . . . . . .
79
4.2 Nonholonomic
systems.
. . . . . . . . .
81
4.2.1
The
unicycle and the nonholonomic integrator
83
4.3 Stabilizing an inverted pendulum . . . . . . . .
89
5
Systems
with
Sensor
or
Actuator
Constraints
93
5.1
The
bang-bang principle of time-optimal control
93
5.2 Hybrid
output
feedback . . . . . . . . . . . .
96
5.3 Hybrid control of systems with quantization 100
5.3.1 Quantizers 100
5.3.2 Static
state
quantization.
. 103
5.3.3 Dynamic
state
quantization 108
5.3.4
Input
quantization . . . . .
116
5.3.5
Output
quantization . . . .
121
5.3.6 Active probing for information 124
6
Systems
with
Large
Modeling
Uncertainty
6.1 Introductory remarks .
.129
129
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